microstructural characterization of yttria-stabilized zirconia plasma-sprayed deposits using...

Acta mater. 49 (2001) 1661–1675www.elsevier.com/locate/actamat

MICROSTRUCTURAL CHARACTERIZATION OF YTTRIA-STABILIZED ZIRCONIA PLASMA-SPRAYED DEPOSITS USING

MULTIPLE SMALL-ANGLE NEUTRON SCATTERING

A. J. ALLEN1†, J. ILAVSKY1, 2, G. G. LONG1, J. S. WALLACE1, C. C. BERNDT3 andH. HERMAN3

1Ceramics Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA,2Department of Chemical Engineering, University of Maryland, College Park, MD 20742, USA and

3Department of Materials Science and Engineering, State University of New York, Stony Brook, NY11794, USA

( Received 9 February 2000; received in revised form 27 October 2000; accepted 31 October 2000 )

Abstract—Density, electron microscopy, elastic modulus, and small-angle neutron scattering studies are usedto characterize the microstructures of yttria-stabilized zirconia plasma-sprayed deposits as a function of bothfeedstock morphology and annealing. In particular, anisotropic multiple small-angle neutron scattering dataare combined with anisotropic Porod scattering results to quantify each of the three main porous componentsin these thermal barrier coating materials: intrasplat cracks, intersplat lamellar pores and globular pores. Aninverse correlation between the volume of porosity and its surface area is confirmed for the as-sprayeddeposits, as is a preferential annealing of intrasplat cracks at elevated temperatures. The average elasticmodulus is correlated with the total void surface area while the elastic anisotropy is related more closely tothe intersplat porosity. However, depending on the feedstock morphology, globular pores are also shown toplay a surprisingly significant role in post-anneal deposit microstructures and properties. 2001 ActaMaterialia Inc. Published by Elsevier Science Ltd. All rights reserved.

Keywords: Plasma spray; Coating; Neutron scattering; Microstructure; Elastic modulus

1. INTRODUCTION

Plasma-sprayed ceramic deposits are widely appliedin engineering practice as thermal barrier coatings(TBC) on substrates or as free-standing parts follow-ing their removal from substrates [1–4]. However,TBCs have engineering reliability problems that maylimit the trustworthiness and value of the systems intowhich they are incorporated [1,5]. An incompleteunderstanding of the complex processing–microstruc-ture–property relationships in plasma-spraying,together with an insufficient microstructural charac-terization of the spray deposits themselves, lies at theheart of these reliability problems. Much currentapplication development is empirically-based withonly a limited use of mathematical models [6].

Plasma-sprayed ceramic deposits possess a compli-cated microstructure formed by the successiveimpacts of semi-molten powder particles on a sub-strate [7]. On impact, they spread out and solidifyrapidly into lamellar splats with cooling rates in the

† To whom all correspondence should be addressed. Tel.:�1-301-975-5982; fax: �1-301-975-5334.

E-mail address: [email protected] (A.J. Allen)

1359-6454/01/$20.00 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.PII: S13 59-6454( 00 )0 0393-1

range 104–105 K s�1. This results in the build-up ofhigh cooling stresses within the splats and the fre-quent appearance of metastable phases [8]. Additionalstresses may arise because of the different thermalexpansion coefficients of the deposit and the sub-strate. Frequently, such stresses are relieved by crack-ing or sliding of the splats. Elevated temperatures dur-ing service life cause further microstructural changesand long-term phase changes [9, 10].

Plasma-sprayed ceramic deposit microstructuresare dominated by two anisotropic distributions ofpreferentially-oriented voids: intersplat lamellar poresand intrasplat cracks. There is also a broad size distri-bution of rounded globular pores [11]. Intersplatpores are mostly parallel to the substrate(perpendicular to the spray direction) whereas intras-plat cracks are mostly perpendicular to the substrate.Variability in these coexisting anisotropic distri-butions, and the closed nature of many of the voids,have limited the applicability of techniques such asimage analysis [12] or mercury intrusion porosimetry[13]. However, the high penetrability of neutronbeams within condensed matter has been exploited tocharacterize the representative microstructures ofboth the open and closed voids in plasma-sprayed

1662 ALLEN et al.: CHARACTERIZATION OF ZIRCONIA DEPOSITS

deposits. Our previous work [14–22] has applied thesmall-angle neutron scattering (SANS) method ofPorod surface characterization to differentiatebetween the surface area orientation distributions ofthe two anisotropic void populations.

In the present paper, we describe how anisotropicmultiple small-angle neutron scattering (MSANS)studies can be used to determine the representativedimensions, porosities, and orientation distributionsof the microstructural void components in plasma-sprayed ceramic deposits. This is done in some detailbecause the application of MSANS analysis to aniso-tropic multicomponent microstructures has not beenpresented previously. The results of the MSANSanalysis are compared with electron microscopyobservations. When sufficient void anisotropy exists,a combination of anisotropic MSANS studies andSANS Porod surface area characterization, togetherwith density measurements, makes feasible the separ-ate parameterization of all three void populationsincluding the globular pores. Using these methods toquantify the microstructural changes that arise fromannealing, new insights are gained into the changesobserved in the properties of annealed plasma-sprayed ceramic deposits such as those revealed byelastic modulus measurement. Finally, we discusshow modulus measurements can be combined withSANS and MSANS studies to provide a more com-plete characterization of the microstructural changesthat typically occur at elevated temperatures duringthe early service life of plasma-sprayed ceramicTBC deposits.

2. EXPERIMENTAL PROCEDURE

2.1. Samples studied

Two yttria-partially-stabilized zirconia feedstockpowder materials were studied, each with 7–8% yttriamass fraction to suppress transformations between thetetragonal zirconia phase and the low-density mono-clinic phase during formation and subsequent thermalcycling of the plasma-sprayed deposits. One, preparedby the fused-and-crushed method and denoted FC(Amdry 142 from Sulzer Plasma Technik Inc., Troy,MI†), contained dense irregular grains in the sizerange 40–115 µm. The other, prepared by a pro-prietary (HOSP) plasma-spheroidization process anddenoted PS (SX233 from Osram Sylvania Inc., Dan-vers, MA), contained less dense globular particles inthe size range 25–95 µm. The lower density was dueto the existence of internal porosity within the PS par-ticles.

The plasma-sprayed deposits were fabricated usinga gas-stabilized spray system (F4� system, Sulzer

† Information on commercial products is given for com-pleteness and does not necessarily constitute or imply theirendorsement by the National Institute of Standards andTechnology.

Plasma Technik Inc., Troy, MI) at the Center forThermal Spray Research, State University of NewYork, Stony Brook, NY. The spray nozzle diameterwas 8 mm, the powder injector diameter, 1.8 mm, andthe gun power was 500 A at 68 V. The powder feedrate was 1.56 kg h�1 (26 g min�1), the argon primarygas flow rate, 2.4 m3 h�1 (40 slpm), the hydrogensecondary gas flow rate, 0.60 m3 h�1 (10 slpm), andthe argon carrier gas flow rate was 0.18 m3 h�1 (3slpm). The spray distances between the torch nozzleand substrate were 90 mm for FC and 65 mm for PS,chosen because these spray distances gave the leastdissimilar total porosities.

Deposits, �5 mm thick, were sprayed onto mildsteel substrates covered with a thin (�0.1 mm) layerof arc-sprayed aluminum. The aluminum layer wasthen dissolved in 20% HCl acid to obtain free-stand-ing deposits. The samples were sectioned to finaldimensions of around 5 mm�25 mm�3.8 mm usinga low-speed diamond saw. Five FC and five PSsamples were prepared for study in the as-sprayedcondition, and also after annealing for 1 h at 1100°C,1200°C, 1300°C or 1400°C. Each annealed samplewas heated to and cooled from its furnace annealingtemperature at a rate set to 600 K h�1 but coolingwas slower than this below 400°C.

The sample densities were determined geometri-cally after precision diamond cutting and grindinginto tetragonal shapes. The precise orthogonal dimen-sions were measured by micrometer and each samplecarefully weighed in air. From the densities given bythe mass-over-volume ratios, total porosities, �T,were deduced to within a standard deviation of±0.1% by volume, based on the average of five inde-pendent density determinations in each case and anassumed theoretical density of 6.00 g cm�3 [8].Accurate evaluation of �T, shown in Table 1, wasessential for reliable microstructural characterizationof the deposits by small-angle scattering.

2.2. Porod (surface) small-angle neutron scatteringcharacterization

All of the small-angle neutron scattering (SANS)measurements were made on the 8-meter SANS

Table 1. Density, total porosity and total surface area resultsa

Total surfaceFeedstock and Density Total porosity,

area, ST (m2

condition (% TD) �T (%)cm-3)

FCAs-sprayed 87.3 (1) 12.7 (1) 3.03 (6)1100°C/1 h 87.6 (1) 12.4 (1) 2.42 (5)1200°C/1 h 88.2 (1) 11.8 (1) 2.03 (4)1300°C/1 h 88.6 (1) 11.4 (1) 1.67 (3)1400°C/1 h 89.8 (1) 10.2 (1) 1.17 (2)PSAs-sprayed 82.9 (1) 17.1 (1) 2.62 (4)1100°C/1 h 84.1 (1) 15.9 (1) 2.16 (3)1200°C/1 h 84.4 (1) 15.6 (1) 1.59 (2)1300°C/1 h 85.3 (1) 14.7 (1) 1.36 (2)1400°C/1 h 85.4 (1) 14.6 (1) 1.08 (2)

a Standard uncertainties in least significant digits given in parentheses.

1663ALLEN et al.: CHARACTERIZATION OF ZIRCONIA DEPOSITS

Fig. 1. Schematic for SANS experiments with the two sampleorientations used. The spray direction, X, is given for the twosample orientations. The direction of Q is given for the one

scattered beam direction shown.

instrument [23] at the Center for Neutron Research,National Institute of Standards and Technology, Gai-thersburg, MD. A monochromatic, collimated beamof neutrons with wavelength, l, controlled by a neu-tron velocity selector, was passed through each paral-lel-sided sample. Scattered neutrons were recordedwith a two-dimensional area detector. An experi-mental schematic is shown in Fig. 1 for the two dif-ferent sample orientations used.

In plasma-sprayed ceramic deposits, the scat-tering arises from a difference, ��r�, in the neutronscattering-length density, a fundamental measure ofthe neutron interaction with matter, at the interfacebetween the voids and the solid splats. The absol-ute-calibrated scattered intensity or macroscopicdifferential scattering cross-section [24, 25],d�(Q)/d�, is a function of the magnitude Q ( [4p/l]sinq, where 2q is the scattering angle) and

direction of the scattering vector, Q. Fig. 1 showsthat Q lies in the scattering plane and is almostorthogonal to both the incident and small-anglescattered beams. In principle, d�(Q)/d� is relatedto the porosity or volume fraction, �, of the scat-tering voids, to their scattering contrast, ��r�2, withthe surrounding material, and to their size, orien-tation and surface-area distributions, projectedalong Q. In practice, the coarse and concentrateddeposit microstructures preclude conventionalSANS studies except for measuring the void sur-face areas from the Porod scattering at high Qwhere Q�3 and � is the opening dimension ofthe voids.

Porod scattering [24] measurements were made todetermine the total surface area per unit volume,ST, of all of the open and closed voids in the samplemicrostructure. Symmetry in the microstructureabout the spray direction enabled a complete aniso-tropic analysis to be performed from one Porodmeasurement on each sample oriented with thespray direction perpendicular to the incident neutronbeam [17, 22]. The beam diameter was 2.4 mm,l 0.6 nm, the wavelength resolution, �l/l

15%, and the sample-to-detector distance was 3.6m, giving a measured Q range of 0.1–1.6 nm�1. Theanisotropic scattered data were corrected for para-sitic background scattering effects, calibratedagainst a silica scattering standard, and sector-aver-aged to give d�(Q)/d� data vs Q in 10° incrementsabout the incident beam position on the instrumentdetector. The apparent Porod surfaces were evalu-ated over a typical Q range of 0.3–1.2 nm�1,depending on the Q direction.

Anisotropic Porod scattering, described in detailelsewhere [17], strongly amplifies the true anisotropyin the surface area orientation distribution of thedeposits. This is important for distinguishing the dif-ferent anisotropic void distributions. For example, fora single population of spheroids, monodispersed inshape, size and orientation, of aspect ratio, b, andorthogonal radii, RO, RO, bRO, the anisotropy in thePorod scattering [26, 27] is proportional to 1/b4. Suchextreme anisotropy in the scattering depends criticallyon the shape and orientation of the scattering voids,and is not easily related to the true pore surface areaorientation distribution. However, an orientationalaverage of the Porod scattering (at given Q) for Qdirections taken over all solid angle with respect tothe sample spray direction, can be used to determinethe total void surface area:

LimitQ→�

�d�(Q)d� �

2p��r�2ST

Q4 (1)

where �d�(Q)/d�� implies an orientational average.As with the total porosity, �T, an accurate value forthe total (open and closed) void surface area withineach sample, ST, determined using equation (1) andshown in Table 1, was found to be essential for anunambiguous microstructural characterization ofeach deposit.

2.3. Multiple small-angle neutron scattering charac-terization

Internal surface area characterization is importantfor quantifying plasma-sprayed microstructures butthe information obtained from Porod scattering aloneis incomplete. For many materials other SANSmethods can complement the surface area informationby providing representative sizes and volume frac-tions of the void component morphologies. Forplasma-sprayed deposits the large void dimensions(0.1–10 µm) cause the Porod scattering to extendthroughout the Q range of most SANS instruments atneutron wavelengths shorter than 0.8 nm. At longerwavelengths the high void concentrations result incopious multiple scattering. Fortunately, in recentyears the MSANS technique has been developed [28–33] and applied to recover some of this structural andvolumetric information [34–36].

The MSANS formalism (see Appendix A for asummary of the algebraic functions used) predicts the


MSANS beam-broadening vs wavelength and wasoriginally developed to extract microstructural para-meters from the broadening associated with concen-trated ensembles of coarse, spherical scatterers [28–30]. It has been progressively enhanced to treat: ran-domly-oriented spheroids [31–33], anisotropically-oriented oblate spheroids (e.g., cracks), and nowmulti-component systems containing two anisotropicdistributions of cracks or planar pores co-existingwith a globular pore population.

Following our previous analysis [28–33] the over-all profile in Q, W(Q�ts), of the neutron beam is con-sidered [28] after it has progressed through a samplethickness, ts. The profile W(Q�ts) is related to themeasured beam-broadening by:

rc limitQ→0

��W�(Q�ts)W(Q�ts)

��1/2

(2)

W�(Q�ts) represents a double-differential with respectto Q and the parameter, rc, is effectively the “radiusof curvature” of the shape of the beam profile in Qat zero Q. Although unmeasureable directly, rc isexperimentally a measure of the width of the beam-broadening and is numerically equal to the standarddeviation of a Gaussian fitted to the beam profile atlow Q. Thus, rc has the same units as Q. The variationof rc with l has provided a more tractable analysisfor obtaining the mean scatterer size than has analysisof the broadened beam profile at just one l value[34–36].

For a microstructure that is axially-symmetricabout the incident beam direction, the scattering iscircularly-symmetric and Moliere’s theory of multiplescattering (in the form of transport equation used byBethe) can be applied to describe the beam profile asit passes through condensed matter. If k 2p/l, thevolume-weighted mean radius of the scatterers isRO, the mean number of multiple scatters within thesample is z̄, and JO(x) denotes a zero order Besselfunction, it has been shown [28–31] that:

W(QRO�ts) k2R2

O

2p �

0

zJO(QROz)exp[�z̄{1�q(z)}]dz

(3)

where z is a dimensionless integration variable. Whilethe integrals must be evaluated numerically, the real-space function, q(z), is related to the single-scatteringintensity by:

q(z) 2p

k2�T�

0

JO(QROz)�d�(Q,nO)d� �

ORIENT

Q dQ

(4)

where, for an axially-symmetric microstructure aboutthe incident beam direction, �T is the total scatteringcross-section per unit sample volume, i.e., �T nsT where sT is the mean total scattering cross-sec-

tion per scatterer, n is the number density of scatterersin the sample, and z̄ �Tts nsTts. The term�d�(Q,nO)/d��ORIENT is the average single-scatterSANS cross-section per unit sample volume takinginto account the axially-symmetric microstructureorientation distribution (see Appendix A). The para-meter, nO, is a phase parameter given by nO 2RO��r�l, and indicates that the multiple scattering

from coarse features is affected both by diffractionand by refraction [28]. Only when nO�1 can refrac-tion effects be ignored.

For a plasma-sprayed ceramic deposit sampleoriented with the spray direction parallel to theincident neutron beam (see Fig. 1), the microstruc-ture is axially-symmetric about the spray directionand the above conditions hold (i.e., the MSANS iscircularly-symmetric). For an orthogonal sampleorientation the anisotropic MSANS data can be cir-cularly-averaged about the incident beam direction.This is equivalent to circularly-averaging the orien-tation distributions of the intrasplat cracks andintersplat pores to give an effective axially-sym-metric microstructure about the incident beam thatis nevertheless not the same as when the sample isoriented with the spray direction parallel to thebeam. Thus, sT, �T, z̄, �d�(Q,nO)/d��ORIENT,q(ζ) and W(QRO�ts) are all different for the twosample orientations.

With known �T, together with known �T and aver-age single-scatter SANS cross-section,�d�(Q,nO)/d��ORIENT, for each of the two sampleorientations used, equations (2)–(4) can be used inprinciple to extract a representative radius, RO, fromthe circularly-averaged MSANS broadening, rc, as afunction of l [28–33] [see Fig. 2(a)]. In practice, afunctional form of �d�(Q,nO)/d��ORIENT must befound that takes into account the anisotropic multi-component void microstructures. In this connection,we assume the intrasplat crack and intersplat porevoid spaces to comprise separate networks of oblatespheroidal volume elements with, respectively, vol-ume-weighted mean radii ROC and ROP, mean aspectratios bC and bP, and porosities �C and �P. Weassume the globular pores to be spheres with meanradius, ROG, and porosity, �G.

For spheroidal elements oriented with their bRO

axes at an angle h with respect to Q, it has beenshown previously for the single-scatter SANS cross-section that [31]:

d�b(Q,nO)X

d� n�fb(Q,nO)X�2 (5a)

and:


Fig. 2. MSANS broadening data (symbols) with model fits(lines) for as-sprayed FC deposit: (a) circularly-averagedMSANS rc vs l for both sample orientations; and (b) aniso-tropic angular variation of MSANS rc at different l, for sampleoriented with the spray direction perpendicular to the incident

beam.

fb(Q,nO)X

ikR2OK(b,X)

1

0

JO(QROK(b,X)x)1 (5b)

�exp�ibnO√1�x2

K(b,X) ��xdxwhere n 3�/4pbR3

O, K(b,X) [1 � (b2�1)X2]1/2,X cos(h), and x is a dimensionless integrationparameter (different from z in the derivation of themultiply-scattered beam profile). Also, the total scat-tering cross-section for spheroid orientation, X, isgiven by [31]:

�b,X 3�

4pbR3O

sb,X where sbX pR2OA(b,X)

(6a)

with

A(b,X) K(b,X)2 �4c2[1�cos(c)�c sin(c)]�

(6b)

and c bnO/K(b,X). With either sample orientation,both d�b(Q,nO)X/d� and �b,X must be averaged overthe separate spheroidal-element orientation distri-butions (with respect to Q) for the intrasplat cracksand intersplat pores, and summed together with thecontribution from the globular pores to give�d�(Q,nO)/d��ORIENT and �T.

To perform the orientational averaging, theapproximate orientation distributions for the cracksand the intersplat pores are separately parameterizedin terms of the probability weights-over-random offinding the short (bRO) axes of their oblate spheroidalelements (the local crack- or pore-normals) within30° of the spray direction, 30–60° from the spraydirection, or 60–90° from it. These weights,pL, pM and pH, respectively, are normalized to unitywith respect to an integration over all possible solidangles for each void system, giving: 0.134pL �0.366pM � 0.500pH 1. (For a random distribution,pL pM pH 1.) For MSANS measurements withthe spray direction parallel to the incident beam, theseweights give the orientation distribution about theincident beam directly. For the circularly-averagedanisotropic MSANS data obtained with the spraydirection perpendicular to the beam, a transformationis necessary to give the probability weights-over-ran-dom of finding the short (bRO) axes in correspondingangular ranges with respect to the incident beamdirection, rather than the spray direction. Equations(5) and (6) can then be numerically-averaged overthese two different axially-symmetric orientation dis-tributions. To orientationally-average the SANScross-sections, an angular mesh size of 0.0025p radi-ans is used for both the azimuthal and polar angleswith respect to the incident beam, and the angle, h,with respect to Q calculated at each orientation. Thedirection of Q is approximated to lie within the planeof the sample.

Equations (3) and (4) must be expressed in termsof a representative microstructural radius, and it wasfound convenient to make this the intrasplat crackradius, ROC. However, the SANS cross-section,�d�(Q,nO)/d��ORIENT in equation (3), combines thescattering from all three void components as do �T

and z̄ in equation (4). Thus, while�d�(Q,nO)/d��ORIENT depends only on ROC as an inde-pendent parameter, it incorporates assumed relativevolume fractions of the three void components, sizeratios (ROP/ROC and ROG/ROC), spheroidal elementaspect ratios (bC and bP) and orientational weights(pL, pM and pH) for each of the crack and intersplatpore networks. Also, �T �C � �P � �G with thesubscripts C, P and G denoting the intrasplat crack,intersplat pore and globular pore components,respectively. Finally, equation (3) must be evaluatedat Q 0 and doubly-differentiated with respect toQ, allowing rc to be obtained at a given l using equ-ation (2).

In principle for a given independently-measured�T, ROC can be determined from the experimentally


measured rc vs l by inversion of equations (2)–(4).Since the crack and intersplat pore networks are suf-ficiently anisotropic and distinct from each other togive significantly different circularly-averagedMSANS broadening for the two sample orientationsstudied, the microstructural parameters can beadjusted until consistency is achieved between theROC values obtained from the MSANS model fittingin the two sample orientations. Unfortunately, theserc vs l variations, alone, are insufficient to determinethe microstructure, uniquely, and two further con-straints must be introduced. One of these is that thecombined surface area of the three void componentsmust be equal to ST, measured independently byPorod scattering. The other constraint is to requireconsistency with the anisotropy actually observed inthe MSANS beam-broadening [shown in Fig. 2(b)]when the sample spray direction is oriented perpen-dicular to the incident neutron beam.

It is not possible to model the anisotropy in theMSANS broadening directly using equations (3) and(4) because of the requirement for axial symmetryabout the incident beam direction. However, for onepopulation of oblate spheroidal elements with orien-tation X, equation (5) predicts [37] that the width ofthe single-scatter profile in Q is proportional to[ROK(b,X)]�1 in the diffraction limit of nO�1. Thismeans that the degree of anisotropy in the scatteringis proportional to 1/b and is (inverse) linearly relatedto the anisotropy in the microstructure. It is assumedthat the anisotropy in the MSANS beam-broadeningfollows this single-scattering anisotropy. TheMSANS anisotropy for each of the two anisotropicvoid populations is calculated by numerically-averag-ing the 1/K(b,X) anisotropy factor over the orientationdistribution with respect to any one direction of Q inthe sample plane. This process is repeated for all azi-muthal Q directions around the incident beam whileapplying the same sector-averaging as used in theanisotropic MSANS data collection. The overall pre-dicted anisotropy is then deduced from a weightedaverage of the sector-averaged component anisotropyfactors for the intrasplat cracks, intersplat pores andglobular pores (where the anisotropy factor is justunity). The weighting is proportional to�C/ROC, �P/ROP and �G/ROG, respectively, to takeinto account both the different component contri-butions to �T and the different single-scatter profilewidths arising from differences amongROC, ROP and ROG. Required consistency between thepredicted and measured MSANS anisotropies wasfound to be the additional constraint needed forobtaining an unambiguous microstructure from theMSANS studies.

In summary, the objective in the anisotropicMSANS analysis is to determine those porosities, �,radii, RO, spheroidal aspect ratios, b, and orientationdistributions, which satisfy the following four con-straints within the experimental uncertainties: (i)component porosities consistent with the total

porosity, �T determined from density measurements;(ii) component surface areas (using standardexpressions for a spheroid) consistent with the totalsurface area, ST, determined from Porod scattering;(iii) fits to the circularly-averaged MSANS rc vs ldata consistent for the two sample orientations used;and (iv) prediction of the MSANS anisotropy consist-ent with that observed in the rc data with the spraydirection perpendicular to the incident beam.

Major approximations in the analysis are that eachanisotropic microstructural void component com-prises oblate spheroidal elements with a constant(modest) aspect ratio, and that each void componentis represented by a volume-weighted mean size. How-ever, since the largest scattering profile width is forQ parallel to the short (bRO) axis of a spheroid, thecontribution to the MSANS broadening from eachanisotropic void component is characterized mainlyby the mean opening dimension, �O.D.�, given by�O.D.� 4bRO/3. Thus, the �O.D.� values for theintrasplat cracks and intersplat pores, together withthe globular pore diameter (2ROG), are the maindimensions of physical significance given by theanalysis. Electron microscopy suggests that, apartfrom occasional coarse features, volume-weightedmean values for these dimensions are representativeof the microstructural void components present.

As a tool in the mathematical analysis, suitablevalues of b were found from experimentation to be0.1 for the spheroidal elements representing the intra-splat cracks and 0.2 for those representing theintersplat pores. These values are much less extremethan the mean macroscopic crack or intersplat–poreaspect ratios, defined as the ratio of the respective�O.D.� to the crack or intersplat–pore “mean pennydiameter” (the large planar dimension of either ofthese void systems traced through any tortuositypresent). The latter aspect ratios cannot be determinedfrom MSANS alone but can be estimated by combin-ing the MSANS analysis with anisotropic elasticmodulus measurement as discussed later.

In the present study, the MSANS broadening wasmeasured for each as-sprayed and annealed ceramicdeposit sample oriented both with the incident beamparallel to the spray direction and perpendicular to it.The beam profile was measured at wavelengths of1.0, 1.2, 1.4, 1.6 and 1.8 nm for the samples and forthe incident beam. These wavelengths were suf-ficiently long, and the samples sufficiently thick, toproduce copious beam-broadening, i.e., z̄ was alwaysgreater than �5. Each rc value was obtained as thestandard deviation of a Gaussian fitted to the beamprofile in the intensity range from �95% of themaximum (at Q 0) down to �40% of themaximum. Previous studies have established that thisfit regime provides reliable determinations of rc fromthe broadening [34–36]. For the spray direction per-pendicular to the incident beam, the anisotropicMSANS was evaluated by sector-averaging the dataon the two-dimensional area detector in 15° sectors


around the incident beam. Given the above con-straints, the normalization requirement for the orien-tational weights, the selected b-values, measured rc

values at five l values for each sample orientation,and the significant MSANS anisotropies observed forthe deposits studied, estimated fractional uncertaintiesin the MSANS-derived component parameters were±10% for the porosities, and ±5% for the surfaceareas, �O.D.�’s, and mean globular pore diameters.

2.4. Electron microscopy

Scanning electron microscopy (SEM) studies werecarried out to provide basic microstructure evaluation,and to validate some of the trends seen in the SANSand MSANS analyses. During polishing, a number ofartifacts (pull-outs) were created that were difficult todistinguish from the voids. Thus, a quantitative imageanalysis was not attempted on these samples.

For each of the 10 samples (see Table 1), twoorthogonal surfaces were prepared for SEM, one par-allel and one perpendicular to the substrate plane. Thesurfaces were prepared by diamond sawing, coarsegrinding with a Grid-Abade disk (TBW Industries,Furlong, PA) to obtain flat surfaces, polishing for 15–20 min with MasterPolish II (Buehler Ltd., LakeBluff, IL) at 250 RPM with a force of approximately200 kPa, and sputter-coated with 15 nm of Au–Pd.With magnifications ranging from 1 kX to 5 kX, anAmray 1830 SEM system (Amray Inc., Bedford, MA)was used to obtain (mainly) back-scattered SEMimages of the void microstructures in the deposits.Some of these are presented in Fig. 3.

2.5. Elastic modulus measurement

The elastic modulus was measured for each sam-ple, both along the spray direction and within the sub-strate plane. These measurements were carried out onthe same sample surfaces studied by SEM, but priorto sputtering the Au–Pd layer. The main purpose wasto correlate the anisotropic mechanical properties ofthe plasma-sprayed deposits with their microstruc-tures, as a function of feedstock morphology andannealing treatment. The present studies weredeveloped from earlier modulus measurements ofsimilar plasma-sprayed ceramic deposits [38, 39].

Each elastic modulus was measured using modifiedcommercial indentation equipment to which load anddisplacement measuring sensors had been added.Indentation loading/unloading curves were recordedto a precision of 3 mN (0.3 g-force) and displace-ments to 10 nm. The polished surfaces of the sampleswere indented with a spherical 2.381 mm (3/32 in.)diameter WC sphere and a nominal 4-N load,resulting in a contact diameter between the WCsphere and the sample of approximately 50 µm anda peak elastic penetration depth of approximately 2.5µm. The apparent elastic modulus of the sample,EAPP, was calculated from the load–displacementcurve using standard Hertzian contact theory [38].The modulus of the sample, ES, could be calculated

from EAPP by taking into account the elastic propertiesof the spherical indenter:

ES 1�m2

S

1EAPP

�(1�m2

I )EI

� (7)

where m is Poisson’s ratio, and the subscripts, I andS, refer to the indenter and sample, respectively. Itwas assumed that mS 0.2, mI 0.22, and EI 614 GPa.

The elastic modulus of the sample was calculatedfrom a least-squares fit of the L2/3 versus h data fromthe loading curve where L was the load and h thedepth of elastic penetration of the spherical indenter.This relationship was linear throughout the loadingrange examined, indicating elastic behavior. Devi-ations from elastic Hertzian contact behavior (e.g.,due to cracking) would have resulted in deviationsfrom linearity. From measurements on standard glasssamples, estimated fractional uncertainties in themeasured moduli were ±0.5%.

Indentation measurements made with the indenterimpinging normal to each deposit top surface wereused to derive an effective Young’s modulus, ES ESPRAY, parallel to the spray direction. Measurements

with the indenter impinging normal to each depositsection cut perpendicular to the substrate were usedto derive an effective Young’s modulus, ES EPLANE, within the substrate plane. The major aim

was to relate the absolute values of ESPRAY andEPLANE, together with the elastic anisotropy given bythe ratio, EPLANE/ESPRAY, to the component voidmicrostructures determined from MSANS. Inaddition, theoretical analyses exist in the literature[40–42], that allowed estimates to be made of themacroscopic “penny” diameters of the intrasplatcracks and intersplat pores by combining the aniso-tropic modulus data with the MSANS-derived para-meters already obtained.

3. RESULTS AND DISCUSSION

3.1. Densities, porosities and surface areas

Table 1 presents the measured percentage theoreti-cal densities (% TD) and derived total porosities forthe 10 plasma-sprayed yttria-stabilized zirconia cer-amic deposit samples. Also listed are the total voidsurface areas, per unit sample volume, determinedfrom an orientational average of the anisotropic Porodscattering over all solid angles, as given by equation(1). The statistical standard uncertainties shown weredetermined as part of the Porod equation fits to thedata in the appropriate Q range. An over (under)-esti-mate of the skeletal density (100% TD) would haveresulted in an over (under)-estimate of the totalporosities, and an under (over)-estimate of the totalsurface areas derived from the Porod scattering. How-


Fig. 3. SEM back-scattered electron images of plasma-sprayed deposit polished sections, oriented with thespray direction vertical, for: as-sprayed deposits, (a) FC and (b) PS; annealed 1100°C/1 h deposits, (c) FC and

(d) PS; annealed 1400°C/1 h deposits, (e) FC and (f) PS.

ever, the value used, 6.00 g cm�3, was deduced fromthe known feedstock composition and agreed with theresults of a recent neutron diffraction study on thesame materials [8].

Table 1 shows that the higher porosity in the sphe-roidized PS feedstock material, compared to that inthe FC feedstock, is reflected in the porosities of theceramic deposits themselves. Even after annealing for1 h at 1400°C, the PS deposit is still more porousthan the FC deposit in the as-sprayed state. In spiteof this, the as-sprayed FC deposit has the greater totalvoid surface area, and its surface area remains slightlyhigher than that of the PS deposit for correspondingannealing treatments. In both systems, the 60%decrease in surface area on annealing is considerablymore marked than the 15–20% fractional decreasein porosity.

The apparent surface area orientation distributionsderived from the Porod scattering strongly amplifythe true microstructural anisotropies [25–27] withinthe deposits and are an important guide to changesduring annealing. For example, in the as-sprayed FCsystem, the intrasplat crack and intersplat pore

components could be discerned clearly, and thecracks were observed to anneal out preferentially dur-ing the heat treatments [22]. These trends can now becompared with the quantitative results derived fromthe MSANS studies.

3.2. Microstructures of the as-sprayed deposits

For the FC feedstock deposit in the as-sprayed con-dition, Fig. 2(a) presents best fits to the MSANS cir-cularly-averaged broadening rc vs l data obtained forthe two sample orientations used (see Fig. 1), subjectto the constraints discussed previously. Figure 2(b)presents corresponding fits, for the several l valuesused, to the anisotropies in rc data obtained with thesample spray direction perpendicular to the incidentneutron beam. Table 2 provides a summary of resultsobtained for the mean crack and pore sizes, as wellas for the porosities and surface areas of each of thethree void components in the FC and PS as-sprayedmicrostructures. The intrasplat crack and intersplatpore �O.D.� values are close to those that would beobtained from the surface-to-volume ratios assumingparallel-sided cracks and pores. The estimated stan-


Table 2. MSANS model fit results for the as-sprayed microstructuresa

Feedstock and void Mean opening dimension,Porosity (%) Surface area (m2 cm-3) Mean pore diameter (µm)

component �O.D.� (µm)

FCIntrasplat cracks 2.9 (5) 1.26 (7) 0.048 (3) –Intersplat pores 6.9 (5) 1.58 (9) 0.096 (6) –Globular pores 2.9 (4) 0.19 (4) – 0.93 (6)PSIntrasplat cracks 4.1 (3) 1.27 (7) 0.068 (3) –Intersplat pores 6.7 (4) 1.09 (5) 0.136 (5) –Globular pores 6.3 (5) 0.26 (4) – 1.53 (6)

a Estimated standard uncertainties in least significant digits given in parentheses.

dard uncertainties in the parameters given in Table 2were deduced by varying the computer model fits tothe MSANS rc vs l data, subject to the previouslydiscussed constraints for �T, ST, and MSANS ani-sotropy [see Fig. 2(b)].

Additional uncertainties arose from the simplifyingassumptions made for the component microstructures.In particular, single volume-weighted mean valueswere assumed for the dimensions of each void type,whereas all of these void components have extendedsize distributions. Generally, the RO values for thespheroidal elements that were used to model the intra-splat cracks and intersplat pores could be set equal toone another, implying that the cracks (b 0.1) haveabout half the opening dimension of the intersplatpores (b 0.2). The mean globular pore radii arecoarser: about 1.3RO for the FC system, and 1.5RO

for the PS system. The derived values for the meanvoid dimensions, given in Table 2, reflect theserelationships. The results in Table 2 suggest that acoarser microstructure of the PS system is responsiblefor a smaller total surface area than in the FC system.The greater total porosity in PS is mainly accountedfor by greater globular porosity.

Although the orientational distributions of theintrasplat cracks and intersplat pores were each rep-resented by just three orientational weights, this wassufficient to model the MSANS anisotropies shownin Fig. 2(b). The orientational weights determined forthe two feedstock systems are given in Table 3.Again, the estimated standard uncertainties weredetermined by varying the MSANS model fits subjectto the constraints. Table 3 indicates that the intrasplatcracks and intersplat pores are more anisotropicallydistributed in the FC than in the PS system.

Table 3. Orientational probability weights-over-random for intrasplat cracks and intersplat pores with respect to the angle (low, medium, high)between their normals and the spray directiona

Feedstock andCrack pL Crack pM Crack pH Pore pL Pore pM Pore pHcondition

FC (all conditions) 0.02 (1) 0.35 (8) 1.74 (8) 3.79 (9) 1.14 (5) 0.15 (4)PS (as-sprayed,

0.05 (3) 0.62 (8) 1.54 (5) 2.88 (6) 1.44 (4) 0.17 (3)1100°C, 1200°C)PS (1300°C) 0.17 (2) 0.77 (3) 1.39 (3) 2.26 (4) 1.25 (3) 0.48 (2)PS (1400°C) 0.18 (3) 0.78 (2) 1.38 (2) 2.15 (3) 1.22 (2) 0.53 (2)

a Note that, for random orientations: pLOW pMED pHIGH 1. Estimated standard uncertainties in least significant digits given in parentheses.

Figure 3 presents SEM images of the depositmicrostructures for the FC and PS deposits, in the as-sprayed condition, and annealed at 1100°C, and1400°C. All of the sample microstructures showedregions of considerable variability, resulting from thestochastic nature of the plasma–spray process. The as-sprayed FC and PS deposit microstructures shown arefairly similar. The mainly horizontal splats are out-lined by the intersplat pores, and some finer, mainlyvertical, intrasplat cracks are visible inside the splats.While intersplat pores coarser than those indicated inTable 2 are seen, the finer ones have opening dimen-sions not inconsistent with the �O.D.� values predictedby MSANS. The intrasplat cracks in the SEM imagesare finer still, and their opening dimensions are alsobroadly consistent with the MSANS predicted�O.D.� values. While it was not possible to discern acoarser PS microstructure here, other differencesbetween the FC and PS deposits, predicted byMSANS, are apparent in the SEM images. Both thecracks and intersplat pores are better aligned in theFC deposit, and the intersplat pores account for alarger fraction of the porosity. Overall, the appearanceof the SEM-imaged microstructures shows qualitativeagreement with the quantitative MSANS analysis.

3.3. Microstructural changes after annealing

Figures 4, 5, and 6 present the results derived froma full MSANS study of all of the samples, measuredusing the two sample orientations. Similar experi-mental uncertainties apply to the data in these figuresas are given in the Tables. Figure 4 presents the dataon the total and void component porosities vs 1-hannealing temperature for the two feedstock systems.Figure 5 presents the corresponding surface area


Fig. 4. Porosities vs annealing temperature for: (a) FC and (b)PS deposits. As-sprayed temperature taken as: 21°C. Lines are

guides to the eye.

changes, and Fig. 6 presents the data on the meanvoid dimensions. (The macroscopic penny diametersof the intrasplat cracks and intersplat pores, shown inFig. 6, were deduced by combining the MSANSanalysis with anisotropic elastic modulus measure-ments and will be discussed later.) At the annealingtemperatures used, both the FC and PS deposits exhi-bit a modest reduction in total porosity, a markedreduction in total void surface area, and a modestcoarsening in most of the microstructure. Also, intras-plat cracks are preferentially annealed out at lowertemperatures than are the intersplat pores, as dis-cussed in previous work [19–22]. However, a com-parison of Figs 4, 5 and 6, with the SEM images ofFig. 3, indicates some striking differences in themicrostructural responses of the FC and PS depositsto annealing.

In the FC deposit, the annealing out of the intras-plat cracks, and the persistence after annealing of theintersplat pores, are evident from their residualporosities and their residual surface areas. The pre-ferred alignments of the intrasplat cracks andintersplat pores were also found to persist (Table 3).

Fig. 5. Surface areas vs annealing temperature for: (a) FC and(b) PS deposits. As-sprayed temperature taken as 21°C. Lines

are guides to the eye.

As shown in Fig. 3(a, c, e), SEM supports theseresults of the MSANS analysis for the FC microstruc-ture annealed for 1 h at 1100°C and 1400°C. Figures4(a), 5(a), and 6(a) suggest that a conversion of theintrasplat cracks to globular or irregular-shaped poresoccurs during annealing, and that the latter coarsenslightly to �1.25 µm. It was not possible to confirmthis trend in the SEM micrographs shown. The globu-lar or irregular pores seen elsewhere in the FC depositshowed a broad size distribution, with many pores inthe �1-µm size regime.

In the PS deposit, fewer intrasplat cracks annealout at lower annealing temperatures (1100°C and1200°C), there is a more significant loss of intersplatpores, and some reduction in the preferred alignmentsof these two void types (Table 3). Figures 4(b), 5(b),and 6(b), suggest a considerable transformation ofintrasplat cracks (and intersplat pores) to globularpores occurs, which coarsen to 3.5 µm at higherannealing temperatures (1300°C and 1400°C). Theseeffects of annealing can be associated with observedreductions in both the magnitude and anisotropy ofthe MSANS broadening of the incident neutron beam,in the absence of proportionate reductions in the totalporosities. The MSANS-based results are also sup-ported by the SEM micrographs for the PS depositshown in Fig. 3(b, d, f), where a broad size distri-


Fig. 6. Mean opening dimensions, �O.D.�, and penny diametersfor intrasplat cracks and intersplat pores, and mean globularpore diameters vs annealing temperature for: (a) FC and (b)PS deposits. As-sprayed temperature taken as 21°C. Lines are

guides to the eye.

bution of globular and irregular-shaped pores areobserved to emerge, especially after the 1400°Canneal. A possible explanation is that the low densityand hollow particles of the PS feedstock cause air orspray gases to become entrapped within the splatsduring spraying. The entrapped gas then leads to thedevelopment and coarsening of the many sphericalpores observed after annealing. Other effects mayinclude the coarse intrasplat crack (associated with aspherical pore) and the fine, unfused particle, bothapparent in Fig. 3(d) for the 1100°C anneal.

By applying Porod scattering theory to the aniso-tropically-oriented spheroidal elements defined above[26, 27], anisotropic Porod surface distributions werederived for the multi-component MSANS modelmicrostructures. For the highly-aligned FC micro-structure, recovery of the observed anisotropic Porodsurface distributions was precluded by the sensitivityof anisotropic Porod scattering to details of the actualshapes of the scattering voids. However, successfulrecovery of the observed anisotropic Porod surfacedistributions was achieved for the less-well aligned

PS microstructure, and is shown in Fig. 7 for the as-sprayed deposit. While this is further validation ofour MSANS approach, it should be noted that Porodscattering studies alone would not have identified thecritically different role of the volumetric globularpores in the two systems, or the different evolutionduring annealing of the relative intrasplat crack andintersplat pore volume fractions. Some annealingeffects are visible in the SEM images, but it was theintroduction of MSANS studies that yielded the stat-istically-representative microstructural parametersrelevant to the prediction of deposit properties.

3.4. Relationship of MSANS-derived microstructureto elastic properties

Figure 8 presents the elastic moduli, ESPRAY andEPLANE, determined from indentation studies on all ofthe deposit samples. The measured moduli are muchless than the Young’s modulus of zirconia, EO 210 GPa [39], due to the void structures within the

deposits. The low value of ESPRAY is attributable tothe population of intersplat pores aligned mainly par-allel to the substrate, while that of EPLANE is attribu-table to the intrasplat crack population (and to mis-aligned intersplat pores). The values of ESPRAY andEPLANE increase as intrasplat cracks and intersplatpores anneal out at elevated temperatures, while theelastic anisotropy, given by the ratio, EPLANE/ESPRAY,decreases as shown in Fig. 9. A comparison of Fig. 9with the earlier figures suggests that, of the MSANS-derived microstructural parameters, the elastic ani-sotropy tracks most closely with the intersplat pores(surface areas recalled in Fig. 9). Thus, these poresshould provide a quantifiable measure of the lamellarpore/splat structures that govern the anisotropicdeposit properties.

The use of anisotropic elastic modulus measure-ment for the microstructural characterization ofplasma-sprayed ceramic deposits has recently beendiscussed [40–42]. To illustrate using the approach of

Fig. 7. Experimental anisotropic Porod surface orientation dis-tribution with MSANS-model prediction for the as-sprayed PS

deposit.


Fig. 8. Indentation elastic moduli along the spray direction,ESPRAY, and within the substrate, EPLANE, vs annealing tempera-ture for: (a) FC and (b) PS deposits. As-sprayed temperature

taken as 21°C. Lines are guides to the eye.

Fig. 9. Elastic anisotropy, EPLANE/ESPRAY, vs annealing tem-perature. Intersplat surface areas shown for comparison. As-sprayed temperature taken as 21°C. Lines are guides to the eye.

Kroupa and Kachanov [41, 42], the intrasplat cracksand intersplat pores are assumed to be aligned perpen-dicular and parallel, respectively, to the substrateplane. The anisotropic moduli, ESPRAY and EPLANE,can then be related separately to EO (above), to mS

(defined earlier), to the overall porosity, �T, and tothe scalar densities, dC and dP, respectively, of the

intrasplat cracks and intersplat pores. These aredefined by: dC �C/(4paC/3) and dP �P/(4paP/3), where �C and �P are the respective

MSANS-derived component porosities, and aC andaP are the macroscopic crack and intersplat–poreaspect ratios. Here, the a-values are assumed to bethe ratio of the appropriate �O.D.� to the penny-shaped crack or pore diameter, in contrast to the b-values of the spheroidal elements defined previously.

From the analysis of the anisotropic elastic moduli[41, 42], aC and aP can be determined, in principle,and used to obtain the penny diameters because the�O.D.� values are already known from the MSANSanalysis. The penny diameters so obtained areincluded in Fig. 6. For the as-sprayed FC and PSdeposits, both the inferred intrasplat–crack andintersplat–pore penny diameters are in the range 6–8µm, acceptably close to the uninterrupted crack andpore lengths seen in Fig. 3(a and b). On annealing,Fig. 6 indicates a marked reduction in the intersplat–pore penny diameters to 1.5–2.0 µm after the 1400°Canneal, while the �O.D.�’s increase slightly as pre-viously discussed. Intersplat pores this short are con-sistent with Fig. 3(f) for the PS deposit. They are notconsistent with Fig. 3(e) for the FC system, but short,partially sintered, intersplat pores were seen else-where in the annealed FC microstructure. This sphe-roidization of intersplat pores during annealing wouldbe one of the expected effects of sintering. In theannealed PS deposits, the intrasplat cracks sphe-roidize in a broadly similar manner but, in theannealed FC deposits, the penny diameters deducedfrom this analysis actually get larger—a result notconfirmed by SEM or by any other result. Theassumptions underlying this analysis break down herebecause misaligned intersplat pores, rather than thelargely annealed-out cracks, govern the value ofEPLANE in the annealed FC deposits.

Despite the differences in the FC and PS micro-structures, the modulus variations show some univer-sal behavior. Figure 10 is a plot of the geometric-mean elastic modulus, �E�, given by �E� (EPLANEEPLANEESPRAY)1/3, vs the total surface area per

unit sample volume (from Porod scattering) for all ofthe samples studied. The calculated curve wasdeduced from the combined anisotropic MSANS andelastic modulus analysis. For these two deposit sys-tems, sprayed under the same conditions of tempera-ture and humidity, the mean elastic modulus is mono-tonically related to the total surface area, itselfdominated by the overall concentration of planarvoids.

4. CONCLUSIONS

In this paper, it has been shown how MSANS stud-ies, in combination with Porod scattering, SEM, anddensity determination, can be used to obtain a quanti-tative representation of the different void componentswithin plasma-sprayed ceramic deposits. The distinct


Fig. 10. Mean elastic modulus data, with predicted theoreticalcurve, vs total surface area per unit sample volume, ST, for all

samples studied.

anisotropies of the intrasplat cracks and intersplatpores are the keys to this nondestructive, statistically-representative, microstructure characterization.Absolute porosities and surface areas of the voidcomponents, the mean opening dimensions andglobular-pore diameter, as well as the approximateintrasplat–crack and intersplat–pore orientation distri-butions, have all been extracted from the analysis. Asfar as we are aware, this is the first time that the indi-vidual void components have been characterized inthis way.

The generic differences between FC and PSplasma-sprayed deposit microstructures have beenquantified and correlated with the feedstock mor-phology. FC deposits have a higher density and retainthe splat/intersplat–pore structure to higher annealingtemperatures. PS deposits have a lower density, butalso a lower surface area and a higher elastic modu-lus. Differences in the anisotropic elastic moduli canbe understood in terms of these differences in theplasma-sprayed microstructures, particularly in theintersplat pore systems. Since the annealing tempera-tures here are typical of present and envisagedoperating temperatures for plasma-sprayed ceramicdeposits in TBC applications, these studies indicatethat TBC durability can be enhanced by adjusting thefeedstock morphology to optimize the TBC mechan-ical property evolution during early service life, sub-ject to structure–property relationships like thatshown in Fig. 10.

In addition to the MSANS-derived parameters, acombination of anisotropic MSANS studies and elas-tic modulus measurements can be used to extract thepenny diameters of the intrasplat cracks and intersplatpores. Further research is needed to relate the meas-ured elastic moduli to the component void microstruc-tures, by incorporating more realistic orientation dis-tributions of the intrasplat cracks and intersplat pores[43]. A complete set of representative microstructural

parameters (porosities, surface areas, �O.D.�’s, globu-lar and penny diameters, and orientationdistributions), for the void components of plasma-sprayed ceramic deposits, is highly desirable. Withappropriate assumptions for polydispersity in themicrostructure, these parameters can be used to pre-dict the generic processing–microstructure–propertyrelationships, not only for the mechanical propertiesbut also for other important TBC properties such asthe dielectric permittivity and thermal conductivity.

Acknowledgements—We thank J. Barker and C. J. Glinka ofthe NIST Center for Neutron Research, for scientific and tech-nical support, together with F. Kroupa of the Institute of PlasmaPhysics, Prague, and M. Kachanov of the Department of Mech-anical Engineering, Tufts University, Medford, for valuablediscussions. This research was supported, in part, by theNational Science Foundation MRSEC Program at the StateUniversity of New York at Stony Brook under Grant No.9632570.

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APPENDIX A

Summary of algebraic termsused

�, �T Porosity: � may refer to one voidcomponent depending on the context;�T refers to the total porosity andincludes intrasplat cracks, intersplat

pores and globular pores.�C, �P, �G Component porosities for the intrasplat

cracks (C), intersplat pores (P) andglobular pores (G).

ST Total surface area per unit sample vol-ume within all open and closed voids;includes intrasplat cracks, intersplatpores and globular pores. (Units in 106

m�1 or m2 cm�3.)�� | Difference in neutron scattering-length

density between that in the solid, r, andthat in the pores (zero). (Units in 1014

m�2, i.e., 1014 m m�3).�� 2 Neutron scattering contrast. (Units in

1028 m�4.)2q, l, ts Scattering angle, wavelength (units in

nm) and sample thickness (units in m ormm), respectively.

Q, Q, k Scattering vector Q with magnitude,Q 4(p/l)sinq, and a directionbisecting the incident and scatteredbeams, which each have wavevectors ofmagnitude, k 2p/l. (Units in nm�1.)

d�(Q)/d� Absolute-calibrated single-scatteringintensity or macroscopic differentialscattering cross-section per unit samplevolume. d�(Q)/d� is the probabilityrate, per unit incident neutron flux andper unit sample volume, of scatteringinto unit solid angle, �, about the scat-tered beam direction defined by the mag-nitude and direction of Q. (Units in m�1

sr�1 or cm�1 sr�1.) The functiond�(Q)/d� gives microstructural infor-mation along a particular direction ofQ approximately perpendicular to theincident beam (see Fig. 1).

RO, b Radius (units in µm) and correspondingoblate aspect ratio (b�1) of spheroidalcrack or pore elements with orthogonalradii, RO, RO and bRO. The intrasplatcrack and intersplat pore void spaces areassumed to comprise separate networksof oblate spheroidal elements with meanradius and aspect ratio: ROC and bC forcracks, ROP and bP for intersplat pores.Globular pores are assumed to have theirown mean radius, ROG.

W(Q�ts) Multiply-scattered neutron beam profilein Q emerging from a sample thickness,ts, normalized to the incident beam pro-file at ts 0. Hence, a dimensionlessfunction of Q. The double-differential ofW(Q�ts) in terms of Q is W�(Q�ts) withunits of Q�2 or nm2.

rc Theoretically, the “radius of curvature”of the beam-profile W(Q�ts) in Q atQ 0. (Hence, units of Q or nm�1).Experimentally, the measured beam-broadening in Q (also with units of Q or


nm�1). For a Gaussian function fitted tothe beam profile W(Q�ts) near Q 0,rc is numerically equal to the standarddeviation of the Gaussian.

z̄ The multiple scattering, i.e., the meannumber of scattering events for each neu-tron passing through the sample.

W(QRO�ts) Multiply-scattered circularly-symmetricneutron beam profile in Q emerging froman axially-symmetric microstructure withscattering void size parameter, RO, andsample thickness, ts, normalized to theincident beam profile at ts 0.

z Dimensionless integration parameterused in equation (3) to relate the beamprofile, W(QRO�ts), to the real spacefunction, q(z), which is defined in termsof the single-scattering cross-section perunit sample volume in equation (4).

�d�(Q,nO)/ Circularly-symmetric, absolute-cali-d��ORIENT brated single-scattering intensity or dif-

ferential scattering cross-section (at agiven Q) per unit sample volume for amicrostructure axially-symmetric aboutthe incident beam direction. The orien-tational average is over the orientationdistribution of the scattering voids withrespect to the direction of Q, and is thesame for all axially-symmetric equival-ent directions of Q. The effects of bothdiffraction and refraction are included.(Units in m�1 sr�1.)

�T, �C, �P, Total scattering cross-section per unit�G sample volume: �T for all voids, �C for

intrasplat cracks, �P for intersplat pores,and �G for globular pores, such that �T

�C � �P � �G. (Units in m�1.)sT Average total scattering cross-section per

scattering void (units: 10�28 m2) suchthat z̄ nsTts �Tts, where n is thenumber density of scatterers.

nO Dimensionless parameter for the phaseshift between neutron de Broglie wavespassing through a scattering void andaround it: nO 2RO��r�l. Parameters,nOC, nOP and nOG give the phase shiftsassociated with intrasplat cracks (C),intersplat pores (P) and globular pores(G) where nOC 2ROC��r�l etc.

K(b,X) Orientational factor for a spheroid ori-ented with its bRO axis at an angle h withrespect to Q and given by: K(b,X) [1 � (b2�1)X2]1/2 with X cos(h).

d�b(Q,nO)X SANS cross-section for a spheroid of/d� aspect ratio, b, and orientation, X,

defined above, with scattering form-fac-tor, fb(Q,nO)X.

x Dimensionless integration parameter

used in equation (5b) to definefb(Q,nO)X for single-scattering. (Not tobe confused with z used in the derivationof W(QRO�ts) for multiple scattering.)

�b,X, sb,X Total scattering cross-section per unitsample volume (�b,X) and per scatteringvoid (sb,X) for spheroids with aspectratio, b, and orientation, X, definedabove. sb,X pR2

OA(b,X) where A(b,X) K(β,X){2 � (4/c2)[1�cos(c)�csin(c)]} and c bnO/K(b,X).

pL, pM, pH Probability weights-over-random offinding the short (bRO) axes of the oblatespheroidal elements (the local crack- orpore-normals) within 30° of the plasmaspray direction, 30–60° from the spraydirection, or 60–90° from it. Used toparameterize the separate approximateanisotropic orientation distributions ofthe intrasplat cracks and intersplat pores.

�O.D.� Mean opening dimension for intrasplatcracks or intersplat pores. In each case,�O.D.� 4bRO/3.

Note that, z̄, W(QRO�ts), q(z), �d�(Q,nO)/d��ORIENT,�T, �C, �P, sT, pL, pM, and pH take different valuesfor the two sample orientations shown in Fig. 1. Alsonote that, while sample orientation is defined withrespect to the incident beam, individual spheroidorientation is defined with respect to Q.

EAPP, EI, ES Elastic modulus from indentationmeasurements: measured apparentmodulus, EAPP, known indenter modu-lus, EI, and deduced modulus of sampleperpendicular to the indented surface,ES. (Units in GPa.)

mI, mS Assumed Poisson’s ratios for theindenter (mI) and the sample (mS) inindentation measurements.

EPLANE, Elastic modulus within the substrateESPRAY, EO plane (EPLANE) and along the spray direc-

tion (ESPRAY) of the plasma-sprayeddeposits as deduced from indentationmeasurements. EO is Young’s modulusfor fully-dense zirconia (210 GPa).

aC, aP Mean macroscopic intrasplat–crack (aC)and intersplat–pore (aP) aspect ratios,defined as the ratio of the �O.D.� to themean “penny” diameter for each compo-nent. Penny diameter take as mean largeplanar dimension for each void compo-nent, traced through any tortuositypresent.

dC, dP Scalar densities of intrasplat cracks (dC)and intersplat pores (dP) defined by: dC

�C/(4paC/3) and dP �P/(4paP/3).

microstructural characterization of yttria-stabilized zirconia plasma-sprayed deposits using...

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